1 Table of Fourier and Generalized Fourier Transforms
Fourier transforms are widely employed in physical, chemical, and engineering applications for
harmonic analysis and for processing acquired data, such as spectroscopic data and images.
Applications include astrophysics, electron microscopy, optics, structure determination (for
example, X-ray, neutron, and electron diffraction), chemical Hyperspectral Imaging
(FT-NIR and FT-IR), and many others. Theoretical studies in quantum mechanics
(QM), QCD, QG, AQFT, and quantum theories on a lattice (QTL) also employ Fourier
transforms.
Fourier–Stieltjes transforms and measured groupoid transforms are useful generalizations
of the ordinary Fourier transform, as summarized in the following table.
Fourier Transforms and Generalized FTs
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| f(t) | ℱ{f(t)} = f(x) | Conditions | Explanation | Description |
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| Gaussian
function | Gaussian function | general | In statistics and
spectroscopy | Gaussian
profiles
remain Gaussian
under Fourier
transformation |
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| Lorentzian
function | Exponential-type
transform | general | In spectroscopy | Associated with
exponentially
decaying
time-domain
signals |
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| Step
or rectangular
function | sin(x)∕x-type
function | general | FT
of a rectangular
pulse | Sinc-type
transform |
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| Triangular
function | sin 2(x)∕x2-type
function | general | Transform of a
triangular
profile | Squared-sinc-type
transform |
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| Series
of equidistant
points | Periodic
reciprocal-space
series | general | Ideal periodic
lattice | Used in
diffraction
theory |
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| Lattice of
infinite planes | Series of
equidistant
reciprocal-space
points | general | One-dimensional
reciprocal space | Used in
crystallography
and diffraction
theory |
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| Helix wrapped
on a cylinder | Bessel functions or
Bessel–Fourier
series | general | Physical
crystallography | Experimentally
truncated to a
finite number of
Bessel terms |
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| c | ( )−1c | Convention-dependent | Constant input | Normalization
depends on
Fourier-transform
convention |
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| f(t) | ∫
f(x) t(x) dx | f(t) ∈ L1(G
l),
with Gl a locally
compact groupoid
[1]; the integral is
defined using a left
Haar measure on
Gl | Fourier–Stieltjes
transform | f(x) ∈ C0(Gl) |
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| m(x) | m(t) =
∫
eitx dm(x) | as above | Inverse
Fourier–Stieltjes
transform | m(t) ∈ L1(G
l)
([2], [3]) |
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| m(x) | m(t) =
∫
eitx dm(x) | When Gl = ℝ and
the integral exists | Usual inverse
Fourier
transform | m(t) ∈ ℝ |
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Note. The hat on f(x) and Gl denotes the transformed quantity or, in the latter case, the dual
object.
References
[1] A. Ramsay and M. E. Walter, Fourier–Stieltjes algebras of locally compact groupoids,
J. Functional Anal. 148: 314–367 (1997).
[2] A. L. T. Paterson, The Fourier algebra for locally compact groupoids, Preprint (2001).
[3] A. L. T. Paterson, The Fourier–Stieltjes and Fourier algebras for locally compact
groupoids (2003). Free PDF file download
